Match the following columns (R= radius,k= rad | vedclass

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Question

(A) The radius of gyration $k$ is related to the moment of inertia $I$ by the formula $I = mk^2$,where $m$ is the mass of the body.For each case:$(A)$

Vedclass · Accepted Answer

$(A)-(R), (B)-(P), (C)-(S), (D)-(Q)$

Vedclass · Answer

(A) The radius of gyration $k$ is related to the moment of inertia $I$ by the formula $I = mk^2$,where $m$ is the mass of the body.For each case:$(A)$ Solid sphere about its tangent: $I = \frac{2}{5}mR^2 + mR^2 = \frac{7}{5}mR^2$. Thus,$k = \sqrt{\frac{7}{5}}R$. Matches $(R)$.$(B)$ Ring about its tangent perpendicular to its plane: $I = mR^2 + mR^2 = 2mR^2$. Thus,$k = \sqrt{2}R$. Matches $(P)$.$(C)$ Uniform solid right circular cone about its central axis: $I = \frac{3}{10}mR^2$. Thus,$k = \sqrt{\frac{3}{10}}R$. Matches $(S)$.$(D)$ Uniform disc about its diameter: $I = \frac{1}{2}(\frac{1}{2}mR^2) = \frac{1}{4}mR^2$. Thus,$k = \frac{R}{2}$. Matches $(Q)$.Therefore,the correct matching is $(A)-(R), (B)-(P), (C)-(S), (D)-(Q)$.

Column $I$	Column $II$
$(A)$ 'k' for a solid sphere rotating about its tangent	$(P)$ $\sqrt{2}R$
$(B)$ 'k' for a ring rotating about its tangent perpendicular to its plane	$(Q)$ $\frac{R}{2}$
$(C)$ 'k' for a uniform solid right circular cone rotating about its central axis	$(R)$ $\frac{\sqrt{7}}{\sqrt{5}}R$
$(D)$ 'k' for a uniform disc rotating about its diameter	$(S)$ $\frac{\sqrt{3}}{\sqrt{10}}R$

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